Non-linear Mechanics

Introduction

Nonlinear mechanics (NM) builds upon the engineering mechanics (statics and dynamics) courses as embedded within the BSc curriculum. This MSc course will complete the education in engineering mechanics for a large group of students. Thus, this course should prepare them as good as possible for common problems faced by mechanical engineers without a specialisation in engineering mechanics. On the other hand, a group of students will continue their training in dynamics and/or statics. For this reason, this course serves as a first step into the more advanced engineering mechanics topics.

Given the broad background of the students participating in this course, abstract formulations will be avoided. Moreover, many topics will be presented as introductions, creating awareness among the students.

Detailed content:

• Introduction to nonlinearities in statics and dynamics

• Principle of virtual work

• Principle of minimum potential

• Green-Lagrange strain tensor

• 2nd Piola-Kirchhoff stress tensor and equilibrium equations

• Alternative stress and strain tensors,

• Geometrical nonlinearity (finite rotations, finite deformations, geometric stiffness effects)

• Introduction to nonlinear material models

• Nonlinear elastic models

• Simple plasticity models (elastic-ideally plastic, isotropic and kinematic hardening)

• Residual stresses

• Application to simple truss and beam problems

• 3D plasticity models work (yield surface, flow rule)

• Introduction to nonlinear dynamics

• Studying simple discrete nonlinear systems using the phase plane

• Quantitative analysis of weakly nonlinear single-degree-of freedom systems using general perturbation theory

• Periodic behaviour of simple nonlinear oscillators, limit cycles and softening/hardening responses

• Basics of bifurcation theory and stability for simple nonlinear dynamic systems

• Introduction to the finite element method in elastostatics (strong, weak or variational, and Galerkin forms of the boundary value problem with linearized kinematics)

• The finite element method for large deformation (total Lagrangian framework)

• Computational tools needed to solve the discrete system of equations